We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$. We set, for $i \in \mathbb { N }$: $$\alpha _ { i } = s ^ { - d } \cdot \left\| F _ { i } - X ^ { d } \right\| _ { r , s } ; \quad \beta _ { i } = \left\| 1 - Q _ { i } \right\| _ { r , s } ; \quad \varepsilon _ { i } = s ^ { - d } \cdot \left\| R _ { i } \right\| _ { r , s } .$$
Verify that, for all $i \in \mathbb { N }$, we have the relation: $$\left( 1 - Q _ { i } \right) \cdot R _ { i } = \left( Q _ { i + 1 } - Q _ { i } \right) \cdot F _ { i + 1 } + R _ { i + 1 }$$